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Theorem rngounval2 25528
Description: The value of the unit of a ring. (Contributed by FL, 12-Feb-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
Hypotheses
Ref Expression
rngunval2.1  |-  X  =  ran  ( 1st `  R
)
rngunval2.2  |-  H  =  ( 2nd `  R
)
rngunval2.3  |-  U  =  (GId `  H )
Assertion
Ref Expression
rngounval2  |-  ( R  e.  RingOps  ->  U  =  (
iota_ u  e.  X A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) ) )
Distinct variable groups:    u, H, x    u, X, x    u, R
Allowed substitution hints:    R( x)    U( x, u)

Proof of Theorem rngounval2
StepHypRef Expression
1 rngunval2.1 . . . 4  |-  X  =  ran  ( 1st `  R
)
2 rngunval2.2 . . . . 5  |-  H  =  ( 2nd `  R
)
3 eqid 2296 . . . . 5  |-  ( 1st `  R )  =  ( 1st `  R )
42, 3rngorn1eq 21103 . . . 4  |-  ( R  e.  RingOps  ->  ran  ( 1st `  R )  =  ran  H )
51, 4syl5eq 2340 . . 3  |-  ( R  e.  RingOps  ->  X  =  ran  H )
65raleqdv 2755 . . 3  |-  ( R  e.  RingOps  ->  ( A. x  e.  X  ( (
u H x )  =  x  /\  (
x H u )  =  x )  <->  A. x  e.  ran  H ( ( u H x )  =  x  /\  (
x H u )  =  x ) ) )
75, 6riotaeqbidv 6323 . 2  |-  ( R  e.  RingOps  ->  ( iota_ u  e.  X A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) )  =  ( iota_ u  e.  ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x ) ) )
8 rngunval2.3 . . 3  |-  U  =  (GId `  H )
9 fvex 5555 . . . . 5  |-  ( 2nd `  R )  e.  _V
102, 9eqeltri 2366 . . . 4  |-  H  e. 
_V
11 eqid 2296 . . . . 5  |-  ran  H  =  ran  H
1211gidval 20896 . . . 4  |-  ( H  e.  _V  ->  (GId `  H )  =  (
iota_ u  e.  ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x ) ) )
1310, 12ax-mp 8 . . 3  |-  (GId `  H )  =  (
iota_ u  e.  ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
148, 13eqtri 2316 . 2  |-  U  =  ( iota_ u  e.  ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
157, 14syl6reqr 2347 1  |-  ( R  e.  RingOps  ->  U  =  (
iota_ u  e.  X A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   iota_crio 6313  GIdcgi 20870   RingOpscrngo 21058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-riota 6320  df-gid 20875  df-rngo 21059
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